Problem: The following system of equations is represented by the matrix equation $\text{A}\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right]=\vec{b}$. $\begin{aligned}x+y+7z&=9 \\2x+7y+z&=10 \\3x-4y+6z&=5\end{aligned}$ ${A}=$ $\vec{b}=$ Represent each row and column in the order in which the variables and equations appear.
Solution: The Strategy A system of equations can be represented by a matrix equation $\text{A}\vec{x}=\vec{b}$, where $\text{A}$ is the coefficient matrix, $\vec{x}$ is the variables vector, and $\vec{b}$ is the constants vector. Each row of the matrix equation represents an equation in the system. [I need an explanation, please!] Representing the system of equations as a matrix equation We are given the system of equations: $\begin{aligned}x+y+7z&=9 \\2x+7y+z&=10 \\3x-4y+6z&=5\end{aligned}$ First, let's rewrite this system to show the coefficients of each variable. $\begin{aligned}{1}x+{1}y+{7}z&=9 \\{2}x+{7}y+{1}z&=10 \\{3}x+({-4})y+{6}z&=5\end{aligned}$ Now, the coefficient matrix can be written as follows. $\left[\begin{array} {ccc} {1} & {1} & {7} \\ {2} & {7} & {1} \\ {3} & {-4} & {6} \end{array} \right]$ We can multiply this matrix by a column vector of variables and set it equal to a column vector with the values on the right side of the equations, as follows. $\left[\begin{array} {ccc} {1} & {1} & {7} \\ {2} & {7} & {1} \\ {3} & {-4} & {6}\end{array} \right]\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right] =\left[\begin{array} {ccc} 9 \\ 10 \\ 5 \end{array} \right]$ This is our matrix equation. Summary $\text{A}$ and $\vec{b}$ are shown below. $\text{A}=\left[\begin{array} {ccc} 1 & 1 & 7 \\ 2 & 7 & 1 \\ 3 & -4 & 6 \end{array} \right]~~~~~~~~~~~~ \vec{b}=\left[\begin{array} {ccc} 9 \\ 10 \\ 5 \end{array} \right]$